Modeling Bimodal Discrete Data Using Conway-Maxwell-Poisson Mixture Models

TL;DR

This paper introduces a mixture of Conway-Maxwell-Poisson distributions to effectively model bimodal discrete data, addressing limitations of traditional Poisson models in capturing over- and under-dispersion.

Contribution

It proposes a novel EM-based method for estimating parameters of CMP mixture models, enabling flexible modeling of bimodal and truncated count data.

Findings

Successfully models bimodal rating and count data

Demonstrates improved fit over traditional Poisson models

Provides computational methods for parameter estimation

Abstract

Bimodal truncated count distributions are frequently observed in aggregate survey data and in user ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an additional mode. Modeling bimodal behavior in discrete data is useful for various purposes, from comparing shapes of different samples (or survey questions) to predicting future ratings by new raters. The Poisson distribution is the most common distribution for fitting count data and can be modified to achieve mixtures of truncated Poisson distributions. However, it is suitable only for modeling equi-dispersed distributions and is limited in its ability to capture bimodality. The Conway-Maxwell-Poisson (CMP) distribution is a two-parameter generalization of the Poisson distribution that allows for over- and under-dispersion. In this work, we propose a mixture of CMPs for capturing a wide range of truncated discrete data, which can exhibit unimodal and bimodal behavior. We present methods for estimating the parameters of a mixture of two CMP distributions using an EM approach. Our approach introduces a special two-step optimization within the M step to estimate multiple parameters. We examine computational and theoretical issues. The methods are illustrated for modeling ordered rating data as well as truncated count data, using simulated and real examples.